What is the probability that, amongst five people in a room, three have the same birthday?
I was wondering about this twist on the birthday problem. I am not a major stats guy so I want your help.
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$\begingroup$ Do you mean exactly three have the same birthday, or at least three? (The two values will be very close, of course.) $\endgroup$– Thomas AndrewsCommented Jul 12, 2013 at 14:54
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$\begingroup$ That depends, what is the probability distribution of birthdays? Is it given to you? $\endgroup$– rurouniwallaceCommented Jul 12, 2013 at 14:56
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1 Answer
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Barring leap years and nonuniform birthday distributions, the probability of exactly three having the same birthday is $$\tag1{5\choose 3}\cdot \frac{364^2}{365^4}\approx 0.000075$$ For exactly four, we get $5\cdot \frac{364}{365^4}\approx0.0000001$ and for five $\frac1{365^4}\approx 0.000000000056$, so $(1)$ is also a good approximation for the question about at least three having the same birthday.
answered Jul 12, 2013 at 14:56
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$\begingroup$ That's a curious graph. I mean, what's wrong with May!? (And what's with November, while we're at it?) $\endgroup$– tomaszCommented Jul 12, 2013 at 15:13
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1$\begingroup$ May has one more day than April and June. November has one fewer day than each of October/December. @tomasz $\endgroup$ Commented Jul 12, 2013 at 15:26
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$\begingroup$ @ThomasAndrews: I meant that in a more +9 months way. But I guess that explains May just fine. Still, August/November is a little weird with the around 0.2% jump over July/October. Worse weather in November keeps people home? :) $\endgroup$– tomaszCommented Jul 12, 2013 at 16:48